The free loop space of a closed oriented d-manifold carries algebraic structure that ordinary loop spaces do not. Intersecting two families of loops where they share a basepoint, and concatenating them there, defines a product that lowers degree by d; rotating loops by the circle adds a square-zero operator, and together these make the shifted homology a Batalin-Vilkovisky algebra whose bracket descends to a Lie algebra on equivariant string homology. A student learns how the umkehr map is built from a Thom collapse even though loop spaces have no Poincare duality, how the same product appears as Hochschild cohomology of cochains and as a ring spectrum structure, and how surfaces and graphs organise the higher operations.
Lecture notes by Ralph Cohen and Alexander Voronov from their courses at the 2003 Almeria summer school, written as an introduction to string topology and a status report on it. String topology studi…